Online variable-sized bin packing with conflicts

We study a new kind of online bin packing with conflicts, motivated by a problem arising when scheduling jobs on the Grid. In this bin packing problem, the set of items is given at the beginning, together with a set of conflicts on pairs of items. A conflict on a pair of items implies that they cannot be assigned to a common bin. The online scenario is realized as follows. Variable-sized bins arrive one by one, and items need to be assigned to each bin before the next bin arrives. We analyze the online problem as well as semi-online versions of it, which are the variant where the sizes of the arriving bins are monotonically non-increasing as well as the variant where they are monotonically non-decreasing.     

New lower bounds for bin packing problems with conflicts

The bin packing problem with conflicts (BPC) consists of minimizing the number of bins used to pack a set of items, where some items cannot be packed together in the same bin due to compatibility restrictions. The concepts of dual-feasible functions (DFF) and data-dependent dual-feasible functions (DDFF) have been used in the literature to improve the resolution of several cutting and packing problems. In this paper, we propose a general framework for deriving new DDFF as well as a new concept of generalized data-dependent dual-feasible functions (GDDFF), a conflict generalization of DDFF. The GDDFF take into account the structure of the conflict graph using the techniques of graph triangulation and tree-decomposition. Then we show how these techniques can be used in order to improve the existing lower bounds.     

Three-dimensional bin packing problem with variable bin height

This paper studies a variant of the three-dimensional bin packing problem (3D-BPP), where the bin height can be adjusted to the cartons it packs. The bins and cartons to be packed are assumed rectangular in shape. The cartons are allowed to be rotated into any one of the six positions that keep the carton edges parallel to the bin edges. This greatly increases the difficulty of finding a good solution since the search space expands significantly comparing to the 3D-BPP where the cartons have fixed orientations. A mathematical (mixed integer programming) approach is modified based on [Chen, C. S., Lee, S. M., Shen, Q. S., 1995. An analytical model for the container loading problem. European Journal of Operational Research 80 (1), 68–76] and numerical experiments indicate that the mathematical approach is not suitable for the variable bin height 3D-BPP. A special bin packing algorithm based on packing index is designed to utilize the special problem feature and is used as a building block for a genetic algorithm designed for the 3D-BPP. The paper also investigates the situation where more than one type of bin are used and provides a heuristic for packing a batch of cartons using the genetic algorithm. Numerical experiments show that our proposed method yields quick and satisfactory results when benchmarked against the actual packing practice and the MIP model with the latest version of CPLEX.     

Branch-and-price algorithms for the dual bin packing and maximum cardinality bin packing problem

There appear to be two versions of the Dual Bin Packing problem in the literature. In addition, one of the versions has a counterpart in the cutting stock literature, known as the Skiving Stock Problem. This paper outlines branch-and-price algorithms for both. We introduce combinatorial upper bounds and well-performing heuristics from the literature in the branch-and-price framework. Extensive computational tests indicate that the branch-and-price approach is superior to the existing branch-and-bound procedures, based on combinatorial bounds. The tests illustrate the influence of different problem characteristics on the computation time and the limits of the branch-and-price approach.     

A robust APTAS for the classical bin packing problem

Bin packing is a well studied problem which has many applications. In this paper we design a robust APTAS for the problem. The robust APTAS receives a single input item to be added to the packing at each step. It maintains an approximate solution throughout this process, by slightly adjusting the solution for each new item. At each step, the total size of items which may migrate between bins must be bounded by a constant factor times the size of the new item. We show that such a property cannot be maintained with respect to optimal solutions. A preliminary version of this paper appeared in Proceedings of the 33rd International Colloquium on Automata, Languages and Programming (ICALP2006), part I, pp. 214–225.     

A lexicographic pricer for the fractional bin packing problem

We propose an exact lexicographic dynamic programming pricing algorithm for solving the Fractional Bin Packing Problem with column generation. The new algorithm is designed for generating maximal columns of minimum reduced cost which maximize, lexicographically, one of the measures of maximality we investigate. Extensive computational experiments reveal that a column generation algorithm based on this pricing technique can achieve a substantial reduction in the number of columns and the computing time, also when combined with a classical smoothing technique from the literature.     

A one-dimensional bin packing problem with shelf divisions

Given bins of size B, non-negative values d and Δ, and a list L of items, each item e∈L with size s_e and class c_e, we define a shelf as a subset of items packed inside a bin with total item sizes at most Δ such that all items in this shelf have the same class. Two subsequent shelves must be separated by a shelf division of size d. The size of a shelf is the total size of its items plus the size of the shelf division. The class constrained shelf bin packing problem (CCSBP) is to pack the items of L into the minimum number of bins, such that the items are divided into shelves and the total size of the shelves in a bin is at most B. We present hybrid algorithms based on the First Fit (Decreasing) and Best Fit (Decreasing) algorithms, and an APTAS for the problem CCSBP when the number of different classes is bounded by a constant C.     

Fuzzy bin packing problem

This paper deals with the fuzzy bin packing problem that is a packing problem of non-rigid rectangles into an open rectangular bin. This problem is different from the conventional bin packing problem, which considers only rigid rectangles. The goal of the fuzzy bin packing problem is to minimize both the height of a packing and the extra cost due to the reduction of each piece. The total cost of the problem is represented as the sum of the height cost and the extra cost due to reductions of the pieces, which is called reduction cost. Because the conventional bin packing problem itself is an NP-hard problem, the presented optimization method assumes that an initial packing for non-reduced pieces has already been found. A closed form solution is presented for fuzzy bin packing problems, in which fuzzy numbers are triangular and the reduction cost is given by a quadratic function.     

A note on a selfish bin packing problem

In this paper, we consider a selfish bin packing problem, where each item is a selfish player and wants to minimize its cost. In our new model, if there are k items packed in the same bin, then each item pays a cost 1/k, where k ≥ 1. First we find a Nash Equilibrium (NE) in time O(n log n) within a social cost at most 1.69103OPT + 3, where OPT is the social cost of an optimal packing; where n is the number of items or players; then we give tight bounds for the worst NE on the social cost; finally we show that any feasible packing can be converged to a Nash Equilibrium in O(n ²) steps without increasing the social cost.     

Online algorithms with advice for the dual bin packing problem

This paper studies the problem of maximizing the number of items packed into n bins, known as the dual bin packing problem, in the advice per request model. In general, no online algorithm has a constant competitive ratio for this problem. An online algorithm with 1 bit of advice per request is shown to be 3/2-competitive. Next, for \(0< \varepsilon < 1{/}2\), an online algorithm with advice that is \((1/(1-\varepsilon ))\)-competitive and uses \({O}(1/\varepsilon )\) bits of advice per request is presented.     

Exact algorithms for the bin packing problem with fragile objects

We are given a set of objects, each characterized by a weight and a fragility, and a large number of uncapacitated bins. Our aim is to find the minimum number of bins needed to pack all objects, in such a way that in each bin the sum of the object weights is less than or equal to the smallest fragility of an object in the bin. The problem is known in the literature as the Bin Packing Problem with Fragile Objects, and appears in the telecommunication field, when one has to assign cellular calls to available channels by ensuring that the total noise in a channel does not exceed the noise acceptance limit of a call.We propose a branch-and-bound and several branch-and-price algorithms for the exact solution of the problem, and improve their performance by the use of lower bounds and tailored optimization techniques. In addition we also develop algorithms for the optimal solution of the related knapsack problem with fragile objects. We conduct an extensive computational evaluation on the benchmark set of instances, and show that the proposed algorithms perform very well.     

THE FFD ALGORITHM FOR THE BIN PACKING PROBLEM WITH KERNEL ITEMS

The FFD algorithm is one of the most famous algorithms for the classical bin packing problem. In this paper,some versions of the FFD algorithm are considered in several bin packing problems. Especially,two of them applied to the bin packing problem with kernel items are analyzed. Tight worst-case performance ratios are obtained.     

Approximation algorithms for the oriented two-dimensional bin packing problem

Given a set of rectangular items which may not be rotated and an unlimited number of identical rectangular bins, we consider the problem of packing each item into a bin so that no two items overlap and the number of required bins is minimized. The problem is strongly NP-hard and finds practical applications in cutting and packing. We discuss a simple deterministic approximation algorithm which is used in the initialization of a tabu search approach. We then present a tabu search algorithm and analyze its average performance through extensive computational experiments.     

The stochastic generalized bin packing problem

The Generalized Bin Packing Problem (GBPP) is a recently introduced packing problem where, given a set of bins characterized by volume and cost and a set of items characterized by volume and profit (which also depends on bins), we want to select a subset of items to be loaded into a subset of bins which maximizes the total net profit, while satisfying the volume and bin availability constraints. The total net profit is given by the difference between the total profit of the loaded items and the total cost of the used bins. In this paper, we consider the stochastic version of the GBPP (S-GBPP), where the item profits are random variables to take into account the profit oscillations due to the handling operations for bin loading. The probability distribution of these random variables is assumed to be unknown. By using the asymptotic theory of extreme values a deterministic approximation for the S-GBPP is derived.     

A branch-and-price algorithm for the variable size bin packing problem with minimum filling constraint

In this paper we consider a variation of the bin packing problem in which bins of different types have different costs and capacities. Furthermore, each bin has to be filled at least to a certain level, which depends on its type. We present a set partitioning formulation and an exact optimization algorithm which exploits column generation and specialized heuristics. We compare our algorithm with the general purpose solver ILOG CPLEX, running on two compact ILP formulations and we report on experimental results on instances we have generated from data-sets for the variable size bin packing problem.     

A hybrid heuristic algorithm for the 2D variable-sized bin packing problem

In this paper, we consider the two-dimensional variable-sized bin packing problem (2DVSBPP) with guillotine constraint. 2DVSBPP is a well-known NP-hard optimization problem which has several real applications. A mixed bin packing algorithm (MixPacking) which combines a heuristic packing algorithm with the Best Fit algorithm is proposed to solve the single bin problem, and then a backtracking algorithm which embeds MixPacking is developed to solve the 2DVSBPP. A hybrid heuristic algorithm based on iterative simulated annealing and binary search (named HHA) is then developed to further improve the results of our Backtracking algorithm. Computational experiments on the benchmark instances for 2DVSBPP show that HHA has achieved good results and outperforms existing algorithms.     

An effective heuristic for the two-dimensional irregular bin packing problem

This paper proposes an adaptation, to the two-dimensional irregular bin packing problem of the Djang and Finch heuristic (DJD), originally designed for the one-dimensional bin packing problem. In the two-dimensional case, not only is it the case that the piece’s size is important but its shape also has a significant influence. Therefore, DJD as a selection heuristic has to be paired with a placement heuristic to completely construct a solution to the underlying packing problem. A successful adaptation of the DJD requires a routine to reduce computational costs, which is also proposed and successfully tested in this paper. Results, on a wide variety of instance types with convex polygons, are found to be significantly better than those produced by more conventional selection heuristics.     

New lower bounds for the three-dimensional finite bin packing problem

The three-dimensional finite bin packing problem (3BP) consists of determining the minimum number of large identical three-dimensional rectangular boxes, bins, that are required for allocating without overlapping a given set of three-dimensional rectangular items. The items are allocated into a bin with their edges always parallel or orthogonal to the bin edges. The problem is strongly NP-hard and finds many practical applications. We propose new lower bounds for the problem where the items have a fixed orientation and then we extend these bounds to the more general problem where for each item the subset of rotations by 90° allowed is specified. The proposed lower bounds have been evaluated on different test problems derived from the literature. Computational results show the effectiveness of the new lower bounds.     

New lower bounds for the three-dimensional orthogonal bin packing problem

In this paper, we consider the three-dimensional orthogonal bin packing problem, which is a generalization of the well-known bin packing problem. We present new lower bounds for the problem from a combinatorial point of view and demonstrate that they theoretically dominate all previous results from the literature. The comparison is also done concerning asymptotic worst-case performance ratios. The new lower bounds can be more efficiently computed in polynomial time. In addition, we study the non-oriented model, which allows items to be rotated.     

An asymptotically exact algorithm for the high-multiplicity bin packing problem

The bin packing problem consists of finding the minimum number of bins, of given capacity D, required to pack a set of objects, each having a certain weight. We consider the high-multiplicity version of the problem, in which there are only C different weight values. We show that when C=2 the problem can be solved in time O( log D). For the general case, we give an algorithm which provides a solution requiring at most C−2 bins more than the optimal solution, i.e., an algorithm that is asymptotically exact. For fixed C, the complexity of the algorithm is O(poly( log D)), where poly(·) is a polynomial function not depending on C.